| import Mathlib |
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| namespace OAI |
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| namespace MatrixAllFields |
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| open scoped BigOperators Topology Polynomial |
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| noncomputable section |
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| namespace MatrixMultiplication.FloorExponentialRates |
|
|
| open Filter |
| open scoped BigOperators Topology Classical |
|
|
| def count (rate : ℝ) (m : ℕ) : ℕ := ⌊Real.exp ((m : ℝ) * rate)⌋₊ |
|
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| theorem count_pos {rate : ℝ} (hr : 0 ≤ rate) (m : ℕ) : 0 < count rate m := by |
| have he : (1 : ℝ) ≤ Real.exp ((m : ℝ) * rate) := |
| Real.one_le_exp (mul_nonneg (Nat.cast_nonneg _) hr) |
| exact lt_of_lt_of_le Nat.zero_lt_one (Nat.le_floor (by simpa only [Nat.cast_one] using he)) |
|
|
| theorem log_count_bounds {rate : ℝ} (hr : 0 ≤ rate) (m : ℕ) : |
| (m : ℝ) * rate - Real.log 2 ≤ Real.log (count rate m : ℝ) ∧ |
| Real.log (count rate m : ℝ) ≤ (m : ℝ) * rate := by |
| have hp : (0 : ℝ) < count rate m := Nat.cast_pos.mpr (count_pos hr m) |
| have hone : (1 : ℝ) ≤ count rate m := by exact_mod_cast count_pos hr m |
| have hround := Nat.lt_floor_add_one (Real.exp ((m : ℝ) * rate)) |
| have hhalf : Real.exp ((m : ℝ) * rate) / 2 ≤ (count rate m : ℝ) := by |
| change Real.exp ((m : ℝ) * rate) < (count rate m : ℝ) + 1 at hround |
| linarith |
| constructor |
| · have h := Real.log_le_log (by positivity : 0 < Real.exp ((m : ℝ) * rate) / 2) hhalf |
| simpa only [Real.log_div (Real.exp_ne_zero _) (by norm_num : (2 : ℝ) ≠ 0), |
| Real.log_exp] using h |
| · have h := Real.log_le_log hp (Nat.floor_le (Real.exp_pos _).le) |
| simpa only [Real.log_exp] using h |
|
|
| theorem tendsto_log_count_div_nat {rate : ℝ} (hr : 0 ≤ rate) : |
| Tendsto (fun m : ℕ => Real.log (count rate m : ℝ) / (m : ℝ)) |
| atTop (𝓝 rate) := by |
| have hinv : Tendsto (fun m : ℕ => Real.log 2 / (m : ℝ)) atTop (𝓝 0) := |
| tendsto_const_nhds.div_atTop tendsto_natCast_atTop_atTop |
| have hlo : Tendsto (fun m : ℕ => rate - Real.log 2 / (m : ℝ)) atTop (𝓝 rate) := by |
| simpa only [sub_zero] using tendsto_const_nhds.sub hinv |
| apply tendsto_of_tendsto_of_tendsto_of_le_of_le' hlo tendsto_const_nhds |
| · filter_upwards [eventually_gt_atTop (0 : ℕ)] with m hm |
| have hm' : (0 : ℝ) < m := Nat.cast_pos.mpr hm |
| have h := div_le_div_of_nonneg_right (log_count_bounds hr m).1 hm'.le |
| simpa only [sub_div, mul_div_cancel_left₀ rate hm'.ne'] using h |
| · filter_upwards [eventually_gt_atTop (0 : ℕ)] with m hm |
| have hm' : (0 : ℝ) < m := Nat.cast_pos.mpr hm |
| have h := div_le_div_of_nonneg_right (log_count_bounds hr m).2 hm'.le |
| simpa only [mul_div_cancel_left₀ rate hm'.ne'] using h |
| |
| theorem tendsto_log_prod_count_div_nat {I : Type*} [Fintype I] (rate : I → ℝ) |
| (hr : ∀ i, 0 ≤ rate i) : |
| Tendsto (fun m : ℕ => Real.log ((∏ i, count (rate i) m : ℕ) : ℝ) / (m : ℝ)) |
| atTop (𝓝 (∑ i, rate i)) := by |
| have h := tendsto_finsetSum Finset.univ (fun i _ => tendsto_log_count_div_nat (hr i)) |
| apply h.congr |
| intro m |
| rw [Nat.cast_prod, Real.log_prod |
| (fun i _ => Nat.cast_ne_zero.mpr (Nat.ne_of_gt (count_pos (hr i) m)))] |
| simp only [div_eq_mul_inv, Finset.sum_mul] |
| |
| abbrev Labels {I : Type*} (rate : I → ℝ) (m : ℕ) := ∀ i, Fin (count (rate i) m) |
| |
| theorem card_labels {I : Type*} [Fintype I] (rate : I → ℝ) (m : ℕ) : |
| Fintype.card (Labels rate m) = ∏ i, count (rate i) m := by |
| simp only [Labels, Fintype.card_pi, Fintype.card_fin] |
| |
| theorem card_labels_pos {I : Type*} [Fintype I] (rate : I → ℝ) |
| (hr : ∀ i, 0 ≤ rate i) (m : ℕ) : 0 < Fintype.card (Labels rate m) := by |
| rw [card_labels] |
| exact Finset.prod_pos fun i _ => count_pos (hr i) m |
| |
| theorem tendsto_log_card_labels_div_nat {I : Type*} [Fintype I] (rate : I → ℝ) |
| (hr : ∀ i, 0 ≤ rate i) : |
| Tendsto (fun m : ℕ => Real.log (Fintype.card (Labels rate m) : ℝ) / (m : ℝ)) |
| atTop (𝓝 (∑ i, rate i)) := by |
| simpa only [card_labels] using tendsto_log_prod_count_div_nat rate hr |
| |
| end MatrixMultiplication.FloorExponentialRates |
| |
| end |
| |
| end MatrixAllFields |
| |
| end OAI |
| |
